Mechanics of 21st Century - ICTAM04 Proceedings
XXI ICTAM, 15–21 August 2004, Warsaw, Poland
COMPUTATIONAL HOMOGENISATION OF MICROHETEROGENEOUS MATERIALS
INCLUDING DECOHESION AT FINITE STRAINS
∗
Stefan Loehnert∗ , Peter Wriggers∗
IBNM, University of Hanover, Appelstr. 9A, 30167 Hannover, Germany
Summary In this paper we present some aspects of computational homogenisation procedures of microheterogeneous materials which
can show decohesion in a cohesive zone around the particles. Applications to this are e.g. polymer coatings stiffened with sand. Due
to the decohesion we get £nite deformations and £nite strains within the RVE. The geometrical and material nonlinearities cause the
main dif£culties. The homogenisation procedure leads to an effective stress strain curve for the RVE. Here we set a special focus on
the adaptive numerical model, the statistical testing procedure and the different boundary conditions (pure traction, pure displacement
and natural boundary conditions) applied on the RVE.
HOMOGENISATION AT FINITE STRAINS
The effective material data of a representative volume element (RVE) is obtained from the relation between the effective
strains ε and the effective stresses σ. The effective material tensor E∗ maps the volume average of the strains on the
volume average of the stresses
hσiΩ = E∗ : hεiΩ
(1)
R
1
with h·iΩ = |Ω| Ω · dΩ. Ω is the domain of the regarded RVE. Equation (1) also holds in the nonlinear range where
E∗ depends on the deformation and maybe on the deformation path itself. A requirement for the strain measure in (1) is
linearity in the displacements, also for the geometrically nonlinear theory.
The average strain theorem states that for a perfectly bonded RVE under uniform displacement boundary conditions the
volume average of the deformation is the same as the given deformation on the boundary. Given a constant deformation
gradient on the boundary F it can be shown that
Z
1
[[u]] ⊗ n0 d∂Ω1,2
,
(2)
hF iΩ0 = F +
0
|Ω0 |
∂Ω10 ∩∂Ω20
where [[u]] is the jump within the displacement £eld in case of a not perfectly bonded microstructure, n 0 is the unit normal
vector, and Ω10 and Ω20 are the domains of the different materials in the RVE respectively. In case of a perfectly bonded
microstructure the displacement jumps vanish. Then hF iΩ0 = F , and similarly hḞ iΩ0 = Ḟ .
The average stress theorem states that in the absence of body forces and under a uniform load on the boundary the
volume average stress is the same as the given stress on the boundary. For £nite deformations one has to distinguish
between different con£gurations. Here we restrict ourselves to the mixed con£guration. We assume a constant £rst PiolaKirchhoff stress tensor P on the boundary. Using Cauchy’s theorem and applying the equations of motion one can show
that
Z
1
hP T iΩ0 = P T +
X ⊗ f 0 dΩ0
(3)
|Ω0 |
Ω0
which for zero body forces yields hP iΩ0 = P.
A commonly accepted criterion for the choice of the size of the RVE is Hill’s condition which states that for a perfectly
bonded microstructure and no body forces the microenergy is equal to the macroenergy of the RVE. This leads to the
fact that the RVE has to be small enough such that from the macroscopic point of view the strain and the stress of the
macroscopic body can be assumed to be approximately constant at the location of the RVE. Due to that the RVE can
be regarded as one point of the macroscopic structure. On the other hand the RVE has to be large enough such that the
boundary £eld ¤uctuations are relatively small. For £nite deformations one can show that in case of Dirichlet boundary
conditions and a perfectly bonded microstructure as well as for pure Neumann boundary conditions and no body forces
we have
.
(4)
hP : Ḟ iΩ0 = hP iΩ0 : hḞ iΩ0
MATERIAL MODEL
The microstructure consists of randomly distributed spherical particles embedded in a binding matrix. The delamination
process is restricted to a domain around the particles. This zone is also called “cohesive zone”. Further details about the
cohesive zone approach can be found for example in [6]. The material law chosen for the binding matrix and the inclusions
is a simple compressible Neo-Hooke material with the Lamé parameters µ(m) , λ(m) and µ(p) , λ(p) respectively. The
material model for the cohesive zone is a simple damage model following the suggestion of Zohdi [7]. The undamaged
(cz)
(cz)
material is also a compressible Neo-Hooke material with the material parameters µ 0 and λ0 . The local degradation
Mechanics of 21st Century - ICTAM04 Proceedings
is represented by a variable α with 0 < α ≤ 1 which “weakens” the stiffness of the material. That means, the material
constants become
(cz)
(cz)
µ(cz) = αµ0
and
λ(cz) = αλ0
.
(5)
The local constraint condition from which α can be computed is Ψ(α) = M(α) − K(α) ≤ 0 where M(α) is a scalar
valued term representing the stress state of the material point
µ
¶
q
tr(σ deg )
tr(σ deg )
M(α) = g(σ deg (α)) : g(σ deg (α)) with g(σ deg (α)) = η1
1 + η2 σ deg −
1
(6)
3
3
and η1 and η2 are parameters scaling the isochoric and deviatoric parts of g(σ deg (α)). K(α) is a threshold value which
depends on the damage variable α itself.
K(α) = Φlim + (Φcrit − Φlim ) αP
(7)
Φcrit is the initial threshold value, and Φlim is the threshold value in the limiting case when the material point has degraded
completely (α = 0). Finally P is an exponent which controls the rate of degradation.
NUMERICAL MODEL AND COMPUTATIONAL TESTING
The RVE is chosen to be a cube with a random distribution of inclusions. In three dimensions it is not easy to generate a
mesh of only nicely shaped hexahedra elements for complex geometries. This motivates a different way of discretisation
where we use only linear cube shaped elements which approximate the particle boundaries only roughly. In order to
increase the accuracy of the discretisation we apply non-conforming elements close to the interfaces between the particles
and the cohesive zone and the cohesive zone and the matrix material. An illustration of this discretisation can be seen
in £gure 1. To get a representative material response one has to do statistical tests with different random distributions of
inclusions. Of special interest is the number of particles needed for each test, the re£nement of the mesh, the number of
tests performed and of course the different boundary conditions. The tests done are performed with pure displacement
boundary conditions, pure traction boundary conditions and natural boundary conditions. The necessary re£nement of the
mesh has been tested in a one particle test. The resulting strain energy as a function of the number of degrees of freedom
is shown in £gure 2.
6.16
6.14
cohesive
zone
6.12
strain energy
matrix
6.1
6.08
6.06
particle
6.04
6.02
0
50000
100000
150000
200000
250000
300000
350000
number of degrees of freedom
Figure 1: Discretisation with non-conforming cubic elements
Figure 2: Convergence: overall energy
To £gure out the required number of inclusions we look at the standard deviation of the stress response for multiple tests
with the same number of inclusions. This value does not decrease with the number of tests performed. Only the number
of inclusions in each test has an in¤uence on it. However, to get a statistical representative response it is necessary to
compute the same test many times with a different random distribution of the inclusions. In £gure 3 one can see that a
relatively low number of inclusions is suf£cient.
At last the number of tests which have to be computed to get a statistically representative result is important. It is not
possible to increase the accuracy of the effective response by computing more tests. But at a higher number of tests
performed, the collection of the effective results of each test form a more Gaussian distribution of the effective results.
This is shown in the histograms in £gure 4 for 200 tests performed.
0.0014
45
0.0012
40
35
stdev(hσi)
0.001
30
normal stresses
0.0008
25
0.0006
20
0.0004
15
shear stresses
10
0.0002
5
0
2
4
6
8
10
12
14
16
18
20
0
0.521
0.5215
0.522
0.5225
0.523
0.5235
0.524
0.5245
number of inclusions
Figure 3: Standard deviation of the effective stresses
Figure 4: Histogram for effective Cauchy stress component hσ11 i
Mechanics of 21st Century - ICTAM04 Proceedings
References
[1] Bishop, J.F.W. and Hill, R.: A Theory of the Plastic Distortion of a Polycrystalline Aggregate under Combined Stresses. Phil. Mag., 42, (1951a),
414-427.
[2] Bishop, J.F.W. and Hill, R.: A Theoretical Derivation of the Plastic Properties of a Polycrystalline Face-Centred Metal. Phil. Mag., 42, (1951b),
1298-1307.
[3] Hill, R.: The Elastic Behaviour of a Crystalline Aggregate. In: Proc. Phys. Soc. (Lond.), A65, pages 349-354 (1952).
[4] Reuss, A.: Berechnung der Fliessgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Z. angew. Math. Mech., 9,
(1929), 49-58
[5] Voigt, W.: Über die Beziehung zwischen den beiden Elastizitätskonstanten isotroper Körper. Wied. Ann., 38, (1889), 573-587.
[6] Needleman, A. et al: Matrix reinforcement, and interfacial failure. In: Suresh, S., Mortensen, A. and Needleman, A.: Fundamentals of metal
matrix composites, Butterworth-Heineman publishers (1993).
[7] Zohdi, T; Wriggers, P.: Computational micro-macro material testing. Archives of Computational Methods in Engineering., 8, 2, (2001), 131-228.
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